A distributed system is defined as a collection of independent computers or processors that are connected by an arbitrary interconnection network [1]}, [2]}. The objective of a task assignment [3]}, [1]} in a distributed system is to find an optimal or suboptimal assignment of tasks to processors (or agents), while satisfying temporal and spatial constraints imposed on the system. A task assignment is either preemptive or non-preemptive [5]}. If a task assignment is preemptive, a task reassignment is allowed in such a way that tasks are transferred between processors (or agents) during their execution for improving the system performance [5]}, [7]}. Recent advances in software agent technology [8]}, [9]}, [10]} in distributed systems allow software entities to observe their environment, and cooperate with other entities if necessary to accomplish their goals. Task migration In this paper we use task migration and process migration interchangeably. between software agents intends to improve the system throughput in a distributed system in which the loads incurred by tasks vary over time [10]}. A task reassignment can be achieved by iterative local task swappings, where a task swapping involves task migrations between a pair of agents as a method of local task reassignment [12]}. A subclass of task assignment (or reassignment) problems involves an equal number of tasks and agents, finding a bijective task assignment between tasks and agents in such a way that the total task assignment (or reassignment) benefit is maximized [13]}. Those bijective task assignment problems and their variants appear in a wide variety of areas in computer science and mathematics [14]}, [15]}, [13]}, [17]}. Given \(n\)  tasks and \(n\)  agents whose connections are described by a (connected) network topology, task swapping cost between two agents often relies on a distance in the network topology; the larger the distance between two agents in the network topology, the larger communication delays of migrating tasks caused by the network. Therefore, we need to consider how task swappings are performed on a given network topology for task reassignments. In this type of problems a group theory can be used to represent task reassignments including iterative task swappings. A group-theoretic approach to representing task assignments or reassignments have already been researched [18]}, [19]}. However, little work has done about task swappings for task reassignments between tasks and agents by using group theory. We propose a group-theoretic model of global task reassignments involving \(n\)  tasks and \(n\)  agents by using local task swappings in a distributed network of several well-known topologies.
